2 Specification

3 Description

Householder's factorization with column interchanges is used in the decomposition F=QU, where F is A with its columns permuted, Q is the first r columns of an m by m orthogonal matrix and U is an r by n upper-trapezoidal matrix of rank r. The pseudo-inverse of F is given by X where

X=UTUUT-1QT.

If the matrix is found to be of maximum rank, r=n,
U is a nonsingular n by n upper-triangular matrix and the pseudo-inverse of F simplifies to X=U-1QT. The transpose of the pseudo-inverse of A is overwritten on A.

On exit: AIJMAXi contains the element of largest modulus in the reduced matrix at the ith stage. If r<n, then only the first r+1 elements of AIJMAX have values assigned to them; the remaining elements are unused. The ratio AIJMAX1/AIJMAXr usually gives an indication of the condition number of the original matrix (see Section 8).

On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.

On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

IFAIL=1

Inverse not found, due to an incorrect determination of IRANK (see Section 8).

7 Accuracy

For most matrices the pseudo-inverse is the best possible having regard to the condition of A and the choice of T. Note that only the singular value decomposition method can be relied upon to give maximum accuracy for the precision of computation used and correct determination of the condition of a matrix (see Wilkinson and Reinsch (1971)).

The computed factors Q and U satisfy the relation QU=F+E where

E2<cεA2+η⁢m-rn-r

in which c is a modest function of m and n, η is the value of T, and ε is the machine precision.

8 Further Comments

The time taken by F01BLF is approximately proportional to mnr.

The most difficult practical problem is the determination of the rank of the matrix (see pages 314–315 of Peters and Wilkinson (1970)); only the singular value decomposition method gives a reliable indication of rank deficiency (see pages 134–151 of Wilkinson and Reinsch (1971) and F08KBF (DGESVD)). In F01BLF a tolerance, T, is used to recognize ‘zero’ elements in the remaining matrix at each step in the factorization. The value of T should be set at n times the bound on possible errors in individual elements of the original matrix. If the elements of A vary widely in their orders of magnitude, of course this presents severe difficulties. Sound decisions can only be made by somebody who appreciates the underlying physical problem.

If the condition number of A is 10p we expect to get p figures wrong in the pseudo-inverse. An estimate of the condition number is usually given by AIJMAX1/AIJMAXr.

9 Example

A complete program follows which outputs the maximum of the moduli of the ‘remaining’ elements at each step in the factorization, the rank, as determined by the given value of T, and the transposed pseudo-inverse. Data and results are given for an example which is a 6 by 5 matrix of deficient rank in which the last column is a linear combination of the other four. Using T=119ε (119 is the norm of the matrix) the rank is correctly determined as 4 and the pseudo-inverse is computed to full implementation accuracy.